Binet type formula for Tribonacci sequence with arbitrary initial numbers

This paper presents detailed procedure for determining the formula for calculation Tribonacci sequence numbers with arbitrary initial numbers T-a,T-b,T-c,T-(n). Initial solution is based on the concept of damped oscillations of Lucas type series with initial numbers T-3,T-1,T-3(n). Afterwards coefficient theta(3) has been determined which reduces Lucas type Tribonacci series to Tribonacci sequence T-0,T-0,T-1(n). Determined relation had to be corrected with a phase shift omega(3). With known relations of unitary series T-0,T-0,T-1(n) with remaining two equations of Tribonacci series sequence T-1,T-0,T-0(n) and T-0,T-1,T-0(n), Binet type equation of Tribonacci sequence that has initial numbers T-a,T-b,T-c(n) is obtained. (C) 2018 Published by Elsevier Ltd.